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Periodic solution and asymptotic behavior of the three-dimensional sixth-order Boussinesq equation in shallow water waves

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Abstract

In this study, the surface wave in inviscid fluid was analyzed. Starting from Euler equation and mass conservation equation, coupled with a set of boundary conditions, the equations obtained by dimensionless method contain two parameters: amplitude parameter \(\alpha \) and shallowness parameter \(\beta \). Using double-series perturbation analysis and scale transformation, the (\(2+1\))-dimensional sixth-order Boussinesq equation is derived for the first time. Based on the Lie group analysis, the generator and the single-parameter invariant group of the new Boussinesq equation are obtained. The conservation laws of the equation are given by using the generator and the adjoint equation. Using Hirota’s bilinear method, we obtain the bilinear equation and residual equation. The existence of the residual equation shows the incomplete integrability of the Boussinesq equation. And judgment conditions for linear stability and orbital stability are obtained. The one-soliton solution and two-soliton solutions are obtained from the Hirota bilinear form. With the Riemann theta function \(\theta _3(\phi ,\;q)\), we study the periodic wave solution. When \(q \rightarrow 0\), it is proved that the periodic solution can be reduced to the one-soliton solution in the limit case based on the asymptotic behavior. Finally, using the graph, it is found that the periodic wave solution can be regarded as the parallel superposition of the one-soliton solution, and the effects of amplitude parameter \(\alpha \) and shallowness parameter \(\beta \) on the amplitude of surface wave are analyzed.

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The data sets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Funding

This work was supported by Key Laboratory of Ministry of Education for Coastal Disaster and Protection, Hohai University (No. 202201), and the National Natural Science Foundation of China (No. 41806104, 41906008).

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Authors and Affiliations

  1. College of Hydraulic Science and Engineering, Yangzhou University, Yangzhou, 225009, China

    Baojun Zhao & Jiaxin Wang

  2. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China

    Huanhe Dong & Lei Fu

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  1. Baojun Zhao
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  2. Jiaxin Wang
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Correspondence to Baojun Zhao.

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Zhao, B., Wang, J., Dong, H. et al. Periodic solution and asymptotic behavior of the three-dimensional sixth-order Boussinesq equation in shallow water waves. Nonlinear Dyn 112, 643–659 (2024). https://doi.org/10.1007/s11071-023-09072-8

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